On page 250 of the rulebook (under the Botches section):
If you score at least one success, even if that success is cancelled out and additional 1s remain, it's just a simple failure.
Under the Spending Willpower section (p. 267-268):
A player may spend one of her character's Willpower points to gain an automatic success on a single action. Only one point of Willpower may be used in a single turn in this manner, but the success is guaranteed and may not be cancelled, even by botches.
Under Potence (p 192):
Each dot that the vampire has in Potence
adds one die to all Strength-related dice rolls. Further,
the player can spend one blood point and change his
Potence dice into an equal number of automatic successes
to all Strength-related rolls for the turn.
Another mention of Automatic Successes in the rulebook might help us find our solution:
The True Love Merit (p. 493-494):
This Merit grants you one automatic success on all Willpower
rolls, which can be negated only by a botch die.
This particular variety of automatic success specifies how it may be cancelled. We know now that Willpower-fueled automatic successes may not be cancelled and that True Love-fueled automatic successes may be, but only in one way. To me, this suggests that the automatic successes from Potence may be cancelled out.
From here, we only need to figure out if these dice would also prevent botches from occurring. Back to page 250.
If none of
your dice comes up a success, and one or more dice are
dice showing 1, the roll is a botch. If you score at least
one success, even if that success is canceled out and
additional 1s remain, it’s just a simple failure.
We now need to decide which sentence is more important. The bolded sentence implies that automatic successes cannot prevent a botch, but the italicized sentence's wording implies that they would instead result in a failure. Based on the wording in True Love, that states that automatic successes can be cancelled by botches, I would argue that the italicized sentence is the more important one, and that because so few effects provide automatic successes, that they were not accounted for in this section.
This would make your first example a failure, as the three 1s would cancel out your Potence dice. The second example would then be a success of one.
tl;dr: Automatic successes function as normal successes in a botch, except when otherwise specified.
Hope this helps!